Advanced Shortcuts

                   "I need to figure out what will be left if I take 52 away from 238."

It is worth distinguishing relatively simple shortcuts from those that involve more complex, multi-step, logical reasoning.

Here are links to videos illustrating some simple shortcuts.

There are many others.

I like to think of advanced shortcuts as similar to the kind of multi-step proofs that are done in high school geometry by using deductive reasoning. In this example, Eva finds that the sum of 29 and 27 is 56. And her reasoning proves that 56 is correct because it relies on known facts and sound logic. To me, what is important here is getting experience with reasoning logically. Knowing how to add is useful, but much less important - she can always add with the app on her phone or her watch.

On the pages that follow, by way of example, I outline a sequence of steps by which children can be helped to reason logically in connection with subtraction of multi-digit numbers. Similar methods can be used for addition, multiplication and division. It should be clear that this work will help them to understand the standard algorithms. But that is not the primary purpose. It is designed to help students learn to reason. It is analogous to instruction in geometry. The theorems of geometry (e.g. alternate interior angles are congruent) are not important. Learning to reason logically is the purpose of the instruction.